Question on trivial extension algebras

Question

Given two finite dimensional algebra $A$ and $B$ such that $A$ is Gorenstein and $B$ is not. Can the trivial extension algebras of $A$ and $B$ be isomorphic? See http://www.sciencedirect.com/science/article/pii/0022404984900586 1.3. for the definition. Gorenstein means here that the injective dimension of the regular module is finite as a left and as a right module.

Answer 1

If $S$ is a finite dimensional algebra, and $M$ a finite dimensional $S$-bimodule, and if we construct the algebras $A=S\ltimes M$ and $B=S\ltimes DM$, then the trivial extension algebras $T(A)=A\ltimes DA$ and $T(B)=B\ltimes DB$ are isomorphic. This is straightforward to check ($T(A)$ and $T(B)$ are naturally isomorphic as vector spaces, since both are $S\oplus DS\oplus M\oplus DM$, so you just need to check that this isomorphism preserves multiplication), and in fact it is proved in

Wakamatsu, Takayoshi, Note on trivial extensions of Artin algebras, Commun. Algebra 12, 33-41 (1984). ZBL0537.16008.

that if $T(A)\cong T(B)$ then there are $S$ and $M$ as above.

Take $S$ to be a non-Gorenstein algebra, and $M=S$.

Then $A=S\ltimes S\cong S\otimes_kk[x]/(x^2)$ is not Gorenstein, since a finite $A$-injective resolution of $A$ restricts to $S$ to give a finite $S$-injective resolution of $S\oplus S$, and similarly for a finite $A$-projective resolution of $DA$.

But $B=S\ltimes DS$ is symmetric, and therefore Gorenstein.

According to my calculations, the simplest example (taking $S=k[x,y]/(x^2,y^2,xy)$) produces:

$A=k[x,y,z]/(x^2,y^2,z^2,xy)$

and

$B=k[w,x,y,z]/(w^2,x^2,y^2,z^2,wy,wz,xy,xz,wx-yz).$

Answer 2

This answer is just an addition to Jeremy Rickards answer that is too long for a comment. It gives the quiver and relations for the relevant trivial extension algebras using QPA (over the field GF(3)).

Trivial extension algebra of $A$:

Quiver( ["v1"], [["v1","v1","x"],["v1","v1","y"],["v1","v1","z"],["v1","v1","t\ e_a1_1"],["v1","v1","te_a1_2"]] )

[ (Z(3)^0)*x^2, (Z(3)^0)xy, (Z(3)^0)xte_a1_2, (Z(3)^0)yx, (Z(3)^0)*y^2, (Z(3)^0)yte_a1_1, (Z(3))xte_a1_1+(Z(3)^0)yte_a1_2, (Z(3))xz+(Z(3)^0)zx, (Z(3))yz+(Z(3)^0)zy, (Z(3)^0)*z^2, (Z(3))xte_a1_1+(Z(3)^0)*te_a1_1*x, (Z(3)^0)*te_a1_1*y, (Z(3))zte_a1_1+(Z(3)^0)*te_a1_1*z, (Z(3)^0)*te_a1_1^2, (Z(3)^0)*te_a1_1*te_a1_2, (Z(3)^0)*te_a1_2*x, (Z(3))xte_a1_1+(Z(3)^0)*te_a1_2*y, (Z(3))zte_a1_2+(Z(3)^0)*te_a1_2*z, (Z(3)^0)*te_a1_2*te_a1_1, (Z(3)^0)*te_a1_2^2, (Z(3)^0)xz*te_a1_2, (Z(3)^0)yz*te_a1_1, (Z(3))xz*te_a1_1+(Z(3)^0)yz*te_a1_2 ]

Trivial Extension of B:

Quiver( ["v1"], [["v1","v1","w"],["v1","v1","x"],["v1","v1","y"],["v1","v1","z\ "],["v1","v1","te_a1_1"]] )

[ (Z(3)^0)*w^2, (Z(3)^0)wy, (Z(3)^0)wz, (Z(3))wx+(Z(3)^0)xw, (Z(3)^0)*x^2, (Z(3)^0)xy, (Z(3)^0)xz, (Z(3)^0)yw, (Z(3)^0)yx, (Z(3)^0)*y^2, (Z(3))wx+(Z(3)^0)yz, (Z(3)^0)zw, (Z(3)^0)zx, (Z(3))wx+(Z(3)^0)zy, (Z(3)^0)*z^2, (Z(3))wte_a1_1+(Z(3)^0)*te_a1_1*w, (Z(3))xte_a1_1+(Z(3)^0)*te_a1_1*x, (Z(3))yte_a1_1+(Z(3)^0)*te_a1_1*y, (Z(3))zte_a1_1+(Z(3)^0)*te_a1_1*z, (Z(3)^0)*te_a1_1^2 ]